Automated Game Analysis via Probabilistic Model Checking: a case study

Automated Game Analysis via Probabilistic

Model Checking: a case study

P. Ballarini1, M. Fisher2, M.J. Wooldridge3

Department of Computer Science, University of Liverpool, UK

Abstract

It has been recognised for some time that there are close links between the various logics developedfor the analysis of multi-agent systems and the many game-theoretic models developed for thesame purpose. In this paper, we contribute to this burgeoning body of work by showing how aprobabilistic model checking tool can be used for the automated analysis of game-like multi-agentsystems in which both agents and environments can act with uncertainty. Specifically, we show howa variation of the well-known alternating offers negotiation protocol of Rubinstein can be encodedas a model for the PRISM model checker and its behaviour analysed through automatic verificationof probabilistic CTL’s properties.

Keywords: agents, uncertainty, probabilistic model-checking.

1 Introduction

In game theory, negotiation is thought of as a game where two players bargainover one or several items. The players’ roles are distinct: the seller is willingto get a profit by selling the bargained items whereas the buyer is keen onspending his/her resources (money) to buy them. The utility of each playeressentially depends on the game outcome (i.e. the value at which an agree-ment is reached). A play of such a game (i.e. extensive game) is expressedin term of an history which describes players’ subsequent moves throughout

1 Email: [email protected] Email: [email protected] Email: [email protected]

Electronic Notes in Theoretical Computer Science 149 (2006) 125–137

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negotiation. The possible actions for a bargainer are either accepting themost recent offer or throwing a counter-offer. A player’s strategy determineshis/her next move given a specific history (i.e. in a monetary negotiation astrategy determine also the value of subsequent offers in case of rejection ofthe most recent offer). The possible strategies for a player are also called purestrategies. If uncertainty is accounted for, then probability distributions aregiven to pure strategies and we talk about mixed strategies, hence expectedutilities. Game-theory aims to study properties of the players’ strategies. Astrategy which maximises a player’s utility (independently of the opponent’sone) is said to be a dominant one, whereas a combination of strategies (for thetwo players) forms a Nash equilibrium if no player will get a higher utility byadopting another strategy given that the opponent will stick to the consideredone. Given a game G, questions like, “is there any dominant strategy?”, or,“is there any Nash-equilibrium strategy combination?”, need to be addressedby researchers. In this paper we focus on a variation of the well known Rubin-stein’s alternating offers’ negotiation framework. In particular we will considerplayers adopting mixed strategies, rather than pure ones, and we show how, insuch a case, a probabilistic model-checker can be used as an alternative meansfor the framework analysis. We consider a limited number of strategy profilesand provide a comparative analysis in terms of the corresponding probabilitydistribution over the set of agreements (hence the expected utility). This ap-proach differs from both the simulation and mathematical analysis techniquesusually employed for game analysis. In particular, our approach is automatic,as are simulation techniques, yet covers all possible behaviours of the sys-tem, as do mathematical analysis techniques. Thus, our approach providesan automatic way to analyse an exact computed expectation concerning thepossible system behaviours.

The structure of this paper is as follows. In Section 2, we review theproblem of negotiation, in particular the “alternating offers” protocol, andin Section 3 we outline the model checking approach, specifically probabilisticmodel checking. We bring these two aspects together in Sections 4 and 5,describing the probabilistic model of the bargaining protocol in Section 4 andcarrying out a range of verification experiments on this scenario in Section 5.Finally, in Section 6, we provide brief concluding remarks.

2 The alternating-offers negotiation framework

A number of protocols and associated strategies for automatic negotiation inmulti-agent systems have been developed over the past two decades. Oneof the earliest, and most influential, was the monotonic concession protocol

P. Ballarini et al. / Electronic Notes in Theoretical Computer Science 149 (2006) 125–137126

and Zeuthen strategy adapted by Rosenschein and Zlotkin [10] from previouswork by Harsanyi and Zeuthen. In that work the authors gave the first realevaluation of such negotiation approaches in automated negotiation settings.Most recent work in automated negotiation has focussed on an alternativemodel of negotiation: the alternating offers model proposed by Rubinstein [8].Here we briefly describe it referring the interested reader to the literature formore details. In Rubinstein’s offers model, agents take it in turns to makean action. The action can be either (i) put forward a proposal (offer), or (ii)accept the most recent proposal. We assume just two agents, a and a, andnegotiation takes place in a sequence of rounds, which we will assume areindexed by the natural numbers. Agent a begins, at round 0, by making aproposal x0, which agent a can either accept (A) or reject (R). If the proposalis accepted, then the deal x0 is implemented. Otherwise, negotiation movesto another round, where agent a makes a proposal (counter-offer) and agent achooses to either accept or reject it. An history of such a game is a (possiblyinfinite) sequence (x0, R, x1, R, . . . , xn, . . .) and a terminal history is a finiteone terminated by A, (x0, R, x1, R, . . . , xn, A), with xn being the bargainingoutcome. A player’s utility describes the preference of a player over histories(i.e. over negotiation outcomes) and players aim to maximise their utilities.In Rubinstein’s framework time deadlines are considered for both players (T a

and T a) hence infinite histories are ruled out. Furthermore disagreement isassumed to be the worst possible outcome hence the first player meeting hisdeadline will accept the most recent offer (i.e. histories are all terminal).

Furthermore, in Rubinstein’s alternating offers framework, the tactic forgenerating players’ proposals is a function of time (1). The offer made by agenta to agent a (either the buyer, b or the seller, s) at time time t (0 ≤ t ≤ T a) fallsbetween its Initial-Price (IP a) and its Reserved-Price (RP a) and is defined asfollows:

pta→a =

{IP a+φa(t)(RP a−IP a) for a=b,

RP a+(1−φa(t))(IP a−RP a) for a=s,(1)

Several time-dependent functions can be characterised by means of the Nego-tiation Decision Function (NDF) φa(t), which is as follows:

φa(t) = ka + (1 − ka)(t

T a)

1ψ (2)

where ka determines the price to be offered by agent a in its first proposal.

By varying the value of ψ in (2), different types of tactic can be obtained(see Figure 1 for φb(t)): with ψ < 1 we talk about boulware tactics (offer in-creases/decreases very slowly for most of the time and reaches the RP rapidlyas the time deadline approaches), with ψ = 1, we have linear tactics whereas

P. Ballarini et al. / Electronic Notes in Theoretical Computer Science 149 (2006) 125–137 127

with ψ > 1 we talk about conceder strategies. In the probabilistic variant

0.6 0.8 10.40.2

(Boulware)

t/T

Price (Conceder)

(Linear)

b

bRP

IP

b

Fig. 1. Types of φb(t)

of the Rubinstein’s framework we have realised the will of a player towardsaccepting an offer must be encoded, in terms of a probability function, as partof a his strategy. We will deal with this aspect in Section 4.

3 Probabilistic Model Checking

Model Checking [4] is a well established methodology for testing a system’smodel against properties expressed in terms of some temporal logic formulae.A model checker takes a model M and a formula φ as inputs and returnseither YES if φ is satisfied on all executions through M (i.e. M |= φ) or NOif it is not (i.e. M �|= φ), providing, in such a case, a counter-example of thechecked property.

Model-checking techniques may be classified with respect to the type ofmodel they refer to. In this sense we may distinguish between models whichallow to represent non-determinism without enclosing any “quantification” ofthe uncertainty (i.e. non-probabilistic system), as opposed to models whichincorporate information about the likelihood of possible future evolutions (i.e.probabilistic systems). Non-probabilistic systems can be modelled in termsof Labelled Transition Systems (LTS), essentially state-graphs whose nodesare attached with propositions stating what is true in a state. Linear Time-temporal Logic [9] and its branching-time extension, the Computational TreeLogic [3], allow for the verification of qualitative properties against a LTS(e.g. properties such as “in any possible execution a safe-state is reached atsome point” or “no deadlock-state can be ever reached”). When indicationsabout the likelihood of the system behaviour can be devised, then a proba-bilistic model may be built. Markov processes [5] are a subclass of stochasticprocesses suitable for modelling systems such that the probability of possible


future evolutions depends uniquely on the current state rather than on its pasthistory (i.e. the path which lead to it). A system’s timing is also taken careof with Markov chains models leading to either Discrete time Markov chains(DTMC), for which time is considered as discrete quantity, or Continuoustime Markov chains (CTMC), when time is thought as a continuous one. Theverification of Markov chains via model checking has been widely developedduring last decades, resulting in the characterisation of specific temporal log-ics and verification algorithms: the Probabilistic Computational Tree Logic(PCTL) [6] for verification of DTMC and the Continuous Stochastic Logic(CSL) [2], for CTMC verification. In the following we briefly introduce thebasic for PCTL model checking, which is the verification technique we arereferring to in this work. For a detailed treatment the reader is referred to theliterature.

Definition 3.1 Given a set of atomic propositions AP , a labelled DTMC Mis a tuple (S,P, L) where S is a finite set of states, P : S×S → [0, 1] isthe transition probability matrix such that ∀s ∈ S,

∑s′∈S P(s, s′) = 1 and

L :S → 2AP is a labelling function.

A path in a given a DTMC M = (S,P, L) and its probability measure areformally characterised in the following definitions.

Definition 3.2 A path σ from state s0 is an infinite sequence σ = s0 → s1 →. . .→sn→ . . . such that ∀i∈N, P(si, si+1)>0. Given σ, σ[k] denotes the k-thelement of σ.

The probability measure of a set of infinite paths with common finite prefixσ ↑ n = s0 → . . . → sn is defined as the product of the probability of thetransitions in the prefix σ↑n.

Definition 3.3 Let σ ↑ n = s0 → . . . → sn be a finite path of M. Theprobability measure of the set of (infinite) paths prefixed by σ↑n is

Prob(σ ↑ n) =n−1∏i=0

P(si, si+1)

if n > 0, whereas Prob(σ ↑ n)=1 if n=0.

Definition 3.4 (PCTL syntax) The syntax of PCTL state-formulae (φ)and path-formulae (ϕ) is inductively defined as follows with respect to the


set of atomic propositions AP :

φ := a | tt | ¬φ | φ ∧ φ | P�p(ϕ)

ϕ := φ U≤t φ

where a∈AP , p∈ [0, 1], t∈N∗∪{∞} and �∈{≥, >,≤, <},

The PCTL semantics is as the CTL one except for probabilistic path-formulae.The formula P�p(ϕ) is satisfied in a state s iff the probability measure ofpaths starting at s and satisfying ϕ, denoted Prob(s, ϕ), fulfils the bound� p. Formally:

s |= P�p(φ′ U≤tφ′′) iff Prob(s, (φ′ U≤tφ′′)) � p

where the semantics of (φ′ U≤tφ′′) with respect to a path σ is defined as:

σ |= φ′U≤tφ′′ iff ∃i≤ t : σ[i] |=φ′′ ∧ ∀j <i, σ[j] |=φ′

Essentially PCTL extends CTL’s expressiveness in two ways: by allowing acontinuous path-quantification (i.e. CTL existential and universal path quan-tifiers are replaced by a single continuous quantifier, namely P�p)

4 and byintroducing a discrete time-bounding for Until-formulae. For a complete treat-ment of PCTL model checking we refer the reader to [6].

4 DTMC Model of Rubintein’s Protocol

In this section we describe how we have built the DTMC model for the negoti-ation framework introduced in Section 2. The model has been implemented inthe Reactive Modules formalism of Alur and Henziger [1], the input languagefor the Probabilistic Model-Checker PRISM [7], and it consists of two mod-ules, the Buyer and the Seller, that reproduce the players’ behaviour describedby the UML state-chart diagrams of Figure 2. In essence the buyer and theseller alternatively throw a bid/counter bid then wait for the other player tomake a decision on their offer. If the offer is accepted (i.e. an agreement hasbeen reached) then the purchase takes place and the negotiation is success-fully completed. State-diagrams in Figure 2 refer to a configuration wherethe Buyer starts the bidding process, hence the Seller is initially waiting toreceive the buyer’s first offer.

For a player the decision on the opponent’s offer is a probabilistic one,which depends on the offered amount. For characterising such a probability

4 PCTL is a superset of CTL’s as: E(φUφ)≡P>0(φUφ) and A(φUφ) ≡ P≥1(φUφ)


Fig. 2. Buyer and Seller state-charts

we have introduced the so called Acceptance Probability functions, S AP ()and B AP () respectively, which return a probability value depending on theoffer’s price (x) and time (t). In defining such functions we have identifiedthree relevant aspects which must be accounted for. First, offers ruled outby a player’s reserved price must be given a null probability, given the timedeadline has not been reached. Secondly, since we are assuming disagreementbeing the worst possible outcome, any price offered at reaching of a player’stime deadline will be certainly accepted. Finally, for the sake of symmetry,players should be equally likely to accept offers of “equally utility” (i.e. whosedistance from their respective RP is the same). With that points in mind wehave defined S AP () and B AP () in the following manner:

S AP (x, t) =

⎧⎪⎪⎨⎪⎪⎩

0 if (x≤S RP ) ∧ (t<T s)

1 − S RPx if (x>S RP ) ∧ (t<T s)

1 if (t≥T s)

(3)

B AP (x, t) =

⎧⎪⎪⎨⎪⎪⎩

1 if (x<=0) ∨ (t≥T b)

1+ S RPx−(B RP+S RP ) if (S RP <x<B RP ) ∧ (t<T b)

0 if (x>B RP ) ∧ (t<T b)

(4)

Figure 3 depicts the acceptance probability for both players for offers re-ceived within the time deadline (i.e. the curves in Figure 3 refer to the projec-tions of function 3 and 4 over a generic time instant t, i.e. functions S AP (t)and B AP (t) with t < T s and t < T b) 5. It should be noted that our defini-tion presumes the buyer is aware of the seller’s reserved price (see function 4).Although unrealistic this choice does not affect the aim of our work which isto show that probabilistic model checking can be fruitfully used for the anal-ysis of games. Alternative formulations of S AP () and B AP () which do notrequire breaching players’ privacy are possible.

5 the depicted curves correspond to a setting such that the seller and buyer reserved priceare, respectively, S RP =1000 and S RP =10000.


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

S_RP=1000 B_RP=10000

acce

ptan

ce p

roba

bilit

y -

>

x=offers’ value ->

bid acceptance probability

S_AP(x)B_AP(x)

Fig. 3. Acceptance probability functions: S AP (t), B AP (t)

Piecewise approximation of NDFs. In our model the continuous NDFs(φ(t)) belonging to the family defined in equation 2 are approximated bypiecewise linear functions consisting of two pieces 6 (see Figure 4) 7. The offerfunction has three parameters: the slope of the first piece, the slope of the sec-ond piece and boundary (switch time) between the pieces. The desired setting(boulware/conceder strategy) is chosen through model configuration so thatdifferent strategy profiles (strategy combinations) are verified (i.e. probabilityfor each possible negotiation outcome are derived).

0

100

200

300

400

500

600

700

800

900

1000

0 2 4 6 8 10

Offe

r’s v

alue

Time

Two segments linear NDFs’ approximation

Buyer-Boul(1/500)-Tswitch=8Seller-Boul(1/500)-Tswitch=8

Buyer-Conc(490/1)-Tswitch=2Seller-Conc(490/1)-Tswitch=2

Fig. 4. NDFs’ linear approximation

6 A two piece line is a good approximation for extreme bargaining tactics (i.e. ψ ∼ 0 orψ>>1), which is the type of non-linear tactics we are interested to address in this work. Lessextreme strategies may be better approximated by multi-piece lines, which would requireminimal modifications to our model in order to be coped with.7 the depicted curves correspond to specific settings for the pieces’ slopes and switch-time.


5 Model’s verification

The probabilistic behaviour (i.e. the Acceptance probability functions) en-coded within the DTMC model of the alternating offer framework, resultsin a probability distribution over the set of possible outcomes of the ne-gotiation (i.e. the interval [S RP, B RP ]), hence over a strategy profile’spayoff. For verifying our model we have considered only settings such thatS RP < B RP , in particular we referred most of our analysis to configurationswith S RP = 10000 and S RP = 11000

Once a strategy profile (e.g. Boulware-Boulware or Conceder-Conceder,etc.) has been chosen by configuration, the corresponding probability distri-bution is determined by verifying, with the PRISM tool, the following PCTLformula against the framework model:

P =?(tt U(agreement)∧(purchase=PV AL)) (5)

The above property captures those evolutions for which an agreement over aspecific value (PV AL) is reached, at some point, during the bargaining process.

With the PRISM tool, the verification of (5) for every possible agreementvalue (PV AL∈ [S RP, B RP ]) provide us with the distribution of the probabilityover the set of outcomes, hence the resulting expected utility can be straight-forwardly derived (for the sake of brevity we do not report it in here).

Before starting the verification phase the model needs to be configured. Theconfiguration requires setting a number of parameters amongst which thestrategy combination which we want to study (i.e. the strategy’s slopes andswitch-time for both players’). The possible strategies are clearly infinite,however for our purpose we consider only a limited number of combinationswhich we aim to compare through the results of model verification. In orderto improve model’s efficiency, the accepting interval (i.e. [S RP, B RP ]) is actu-ally set by means of two distinct parameters: the interval width and its offsetfrom the origin. We observe that the numerical result of model verification isaffected by the chosen interval (as a result of functions (3) and (4) definition).

In the following we report some of the results we have obtained by verifica-tion of (5) for an accepting interval arbitrary set to [10000, 11000] (i.e. width:103,offset:104). We will compare the resulting probability distribution (cumula-tive) for different strategy profiles, pointing out which strategy is dominantamongst the considered ones. The results we present are grouped accordingto different strategy types. We denote Lin(x)-Lin(y), non-linear strategy pro-files such that x and y are the slopes of the first segment for, respectively, thebuyer’s and the seller’s NDF and, furthermore, assuming the players’ NDFs


intersect before the switch-time. On the other hand, for example, we useBoul(x/x′)-Conc(y/y′) to denote a profile such that players’ NDFs do notintersect before the switch-time and the buyer is adopting a boulware tac-tic with x and x′ being the slopes, respectively, before and after switching,whereas the seller is using a conceder strategy with slopes y and y′.

Linear-Linear symmetrical: in this setting we consider Lin(x)-Lin(y) pro-files of equal slope (i.e. x = y) . In Figure 5(a) the probability distribution forthe possible agreements is compared for different slope’s value (i.e. Lin(10),Lin(50) and Lin(100)). This graph essentially shows that larger slope’s valuesresult in a higher probability for equal values in the accepting interval. Thecorresponding expectation values are roughly the same: Exp(Lin(10))∼498,Exp(Lin(50)) ∼ 499 and Exp(Lin(100)) ∼ 500, showing a larger slope tendto advantage the seller. This is also confirmed by the graphs in Figure 5(b)that represent the cumulative distribution functions for the three cases. Thecurves in there show that larger strategy’s gradients are better from the sellerpoint of view as they cumulate most of the probability close to the supremumof the accepting interval.

0.0001

0.001

0.01

0.1

1

10000 10100 10200 10300 10400 10500 10600 10700 10800 10900 11000

Acc

epta

nce

Pro

babi

lity

Offer Value

Lin(100)-Lin(100)Lin(50)-Lin(50)Lin(10)-Lin(10)

(a) Probability distribution

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700 800 900 1000

Acc

epta

nce

Cum

ulat

ive

Pro

babi

lity

Offer Value


(b) Cumulative distribution

Fig. 5. Lin(x)-Lin(y) symmetrical profiles

Linear-Linear asymmetrical: here we consider profiles of type Lin(x)-Lin(y) but with different slope’s value (i.e. x �= y). For studying the in-cidence of the strategy’s gradient on the probabilistic outcome of negotia-tion we consider a fixed value for the slope of one player while varying theother’s. In Figure 6 the cumulative distributions for several asymmetricalLin(x)-Lin(y) profiles are depicted. We observe that by increasing the buyer’sslope while the seller’s one is set, for example, to 10 (e.g. by comparing thecurves for profiles Lin(50)-Lin(10), and Lin(100)-Lin(10)) the probabilitytend to cumulate closer to the supremum of the interval (which is good for


the seller). Again this is confirmed by looking at the expectation values,which show the same tendency, with Exp(Lin(50)Lin(10)) ∼ 692, whereasExp(Lin(100)Lin(10))∼804.

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700 800 900 1000

Acc

epta

nce

Cum

ulat

ive

Pro

babi

lity

Offer Value


Lin(10)-Lin(100)Lin(50)-Lin(100)

Fig. 6. Lin(x)-Lin(y) asymmetrical profiles: cumulative probaility

Non-linear asymmetrical: similar conclusions are valid also when we con-sider non-linear profiles for which the NDFs intersect after switching andslopes are different (for both pieces). In Figure 7(a) the cumulative prob-ability for asymmetrical Conceder combinations with different gradients arecompared. There we can observe that profiles which switch to the low gra-dient piece of the strategy earlier tend to cumulate the probability closer tothe higher utility half of the interval. This is evident, for example, if wecompare curves of equal gradients but different swith time as in Conc(10/1)-Conc(100/10)-T sw :4 and Conc(10/1)-Conc(100/10)-T sw :8 in Figure 7(a),the former corresponding to an earlier switch time for the seller (equal to 4)the latter to a delayed switch time (equal to 8).

Similarly, in Figure 7(b) the cumulative probability for Boul(1/100) -Conc(100/1) profiles are compared with respect to to different value of switchtime for the seller. The graphs in Figure 7(b) confirm that for constant slope’svalues, a non-linear strategy with the earliest switch time is dominant.

6 Conclusion

In this paper we have shown a different approach to the verification of multi-player games which can be used as an alternative to (or in conjunction with)analytical methods and/or simulation. We have illustrated how the analysis


0

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0 100 200 300 400 500 600 700 800 900 1000

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Offer Value

Conc(10/1)-Conc(100/10)-Ts_sw:4Conc(10/1)-Conc(100/10)-Ts_sw:8Conc(100/10)-Conc(10/1)-Tb_sw:4Conc(100/10)-Conc(10/1)-Tb_sw:8

(a) Conc-Conc asymmetrical

0

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0 100 200 300 400 500 600 700 800 900 1000

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Boul(1/100)-Conc(100/1)-Tb:20-Ts:9Boul(1/100)-Conc(100/1)-Tb:20-Ts:6Boul(1/100)-Conc(100/1)-Tb:20-Ts:2

(b) Boul-Conc asymmetrical

Fig. 7. Outcome cumulative probability for non-linear strategy profiles

of specific negotiation (mixed) strategies can be performed by means of proba-bilistic model checking. This is achieved by developing an ad hoc probabilisticmodel which is then verified against probabilistic properties with the PRISMmodel checker.

This analysis has helped in comparing the effect that several strategicvariables has on the probabilistic outcome of negotiation. In particular wehave shown that, larger values of the offers function’s slope result in largerexpected agreements as probability distribution for accepted offers tends tocumulate toward the supremum of the accepting interval. Furthermore withrespect to non-linear strategies of constant slopes, the earliest switch time(to the low gradient piece of piecewise linear offer function) is dominant withrespect to the others.

At the best of our knowledge, this work provides a novice contributionfor using a well established and effective automated verification technique asmodel-checking for game analysis. While we have only presented a selectionof the results obtained and much work has still to be done we believe that thisapproach has the potential for improvements in game analysis, in automatedanalysis procedures, and in the development of sophisticated negotiation sce-narios. Specifically our first priority for future developments is to address theissue of Nash equilibria analysis via model-checking.

Acknowledgement: This work was supported by an EC Marie Curie Fel-lowship under contract number HPMF-CT-2001-00065.

References

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[3] E. Clarke, E. Emerson, and A. Sistla. Automatic verification of finite-state concurrentsystems using temporal logic specifications. ACM Transactions on Programming Languagesand Systems, 8(2):244–263, 1986.

[4] E. M. Clarke, O. Grumberg, and D. A. Peled. Model Checking. MIT Press, 2000.

[5] W. Feller. An introduction to probability theory and its applications. John Wiley and Sons,1968.

[6] H. A. Hansson and B. Jonsson. A framework for reasoning about time and reliability. In Proc.10th IEEE Real -Time Systems Symposium, pages 102–111, Santa Monica, Ca., 1989. IEEEComputer Society Press.

[7] M. Kwiatokowska, G. Norman, and D. Parker. Probabilistic symbolic model checking withprism: A hybrid approach. International Journal on Software Tools for Technology Transfer(STTT), 2004.

[8] M. J. Osborne and A. Rubinstein. Bargaining and Markets. The Academic Press: London,England, 1990.

[9] A. Pnueli. A temporal logig of programs. Proc. of the 18th IEEE Symposioum on Foundationsof Computer Science., pages 46–57, 1977.

[10] J. S. Rosenschein and G. Zlotkin. Rules of Encounter: Designing Conventions for AutomatedNegotiation among Computers. The MIT Press: Cambridge, MA, 1994.


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Automated Game Analysis via Probabilistic Model Checking: a case study